1 From the book Networks, Crowds, and Markets: Reasoning about a Highly Connected World. By David Easley and Jon Kleinberg. Cambridge University Press, 010. Complete preprint on-line at Chapter 19 Cascading Behavior in Networks 19.1 Diffusion in Networks A basic issue in the preceding several chapters has been the way in which an individual s choices depend on what other people do this has informed our use of information cascades, network effects, and rich-get-richer dynamics to model the processes by which new ideas and innovations are adopted by a population. When we perform this type of analysis, the underlying social network can be considered at two conceptually very different levels of resolution: one in which we view the network as a relatively amorphous population of individuals, and look at effects in aggregate; and another in which we move closer to the fine structure of the network as a graph, and look at how individuals are influenced by their particular network neighbors. Our focus in these past few chapters has been mainly on the first of these levels of resolution, capturing choices in which each individual is at least implicitly aware of the previous choices made by everyone else, and takes these into account. In the next few chapters, we bring the analysis closer to the detailed network level. What do we gain by considering this second level of resolution, oriented around network structure? To begin with, we can address a number of phenomena that can t be modeled well at the level of homogeneous populations. Many of our interactions with the rest of the world happen at a local, rather than a global, level we often don t care as much about the full population s decisions as about the decisions made by friends and colleagues. For example, in a work setting we may choose technology to be compatible with the people we directly collaborate with, rather than the universally most popular technology. Similarly, we may adopt political views that are aligned with those of our friends, even if they are nationally in the minority. Draft version: June 10,
2 564 CHAPTER 19. CASCADING BEHAVIOR IN NETWORKS In this way, considering individual choices with explicit network structure merges the models of the past several chapters with a distinct line of thinking begun in Chapter 4, when we examined how people link to others who are like them, and in turn can become more similar to their neighbors over time. The framework in Chapter 4 dealt explicitly with network connections, but did not explore the individual decision-making that leads people to become similar to their neighbors: instead, a tendency toward favoring similarity was invoked there as a basic assumption, rather than derived from more fundamental principles. In contrast, the last several chapters have developed principles that show how, at an aggregate population level, becoming similar to one s neighbors can arise from the behavior of individuals who are seeking to maximize their utility in given situations. We saw in fact that there are two distinct kinds of reasons why imitating the behavior of others can be beneficial: informational effects, based on the fact that the choices made by others can provide indirect information about what they know; and direct-benefit effects, in which there are direct payoffs from copying the decisions of others for example, payoffs that arise from using compatible technologies instead of incompatible ones. We now connect these two approaches by exploring some of the decision-making principles that can be used to model individual decision-making in a social network, leading people to align their behaviors with those of their network neighbors. The Diffusion of Innovations. We will consider specifically how new behaviors, practices, opinions, conventions, and technologies spread from person to person through a social network, as people influence their friends to adopt new ideas. Our understanding of how this process works is built on a long history of empirical work in sociology known as the diffusion of innovations [115, 351, 38]. A number of now-classic studies done in the middle of the 0 th century established a basic research strategy for studying the spread of a new technology or idea through a group of people, and analyzing the factors that facilitated or impeded its progress. Some of these early studies focused on cases in which the person-to-person influence was due primarily to informational effects: as people observed the decisions of their network neighbors, it provided indirect information that led them to try the innovation as well. Two of the most influential early pieces of research to capture such informational effects were Ryan and Gross s study of the adoption of hybrid seed corn among farmers in Iowa  and Coleman, Katz, and Menzel s study of the adoption of tetracycline by physicians in the United States . In Ryan and Gross s study, they interviewed farmers to determine how and when they decided to begin using hybrid seed corn; they found that while most of the farmers in their study first learned about hybrid seed corn from salesmen, most were first convinced to try using it based on the experience of neighbors in their community. Coleman, Katz, and Menzel went further when they studied the adoption of a new drug by doctors,
3 19.. MODELING DIFFUSION THROUGH A NETWORK 565 mapping out the social connections among the doctors making decisions about adoption. While these two studies clearly concerned very different communities and very different innovations, they like other important studies of that period shared a number of basic ingredients. In both cases, the novelty and initial lack of understanding of the innovation made it risky to adopt, but it was ultimately highly beneficial; in both cases, the early adopters had certain general characteristics, including higher socio-economic status and a tendency to travel more widely; and in both cases, decisions about adoption were made in the context of a social structure where people could observe what their neighbors, friends, and colleagues were doing. Other important studies in the diffusion of innovations focused on settings in which decisions about adoption were driven primarily by direct-benefit effects rather than informational ones. A long line of diffusion research on communication technologies has explored such direct-benefit effects; the spread of technologies such as the telephone, the fax machine, and has depended on the incentives people have to communicate with friends who have already adopted the technology [16, 85]. As studies of this type began proliferating, researchers started to identify some of the common principles that applied across many different domains. In his influential book on the diffusion of innovations, Everett Rogers gathered together and articulated a number of these principles , including a set of recurring reasons why an innovation can fail to spread through a population, even when it is has significant relative advantage compared to existing practices. In particular, the success of an innovation also depends on its complexity for people to understand and implement; its observability, so that people can become aware that others are using it; its trialability, so that people can mitigate its risks by adopting it gradually and incrementally; and perhaps most crucially, its overall compatibility with the social system that it is entering. Related to this, the principle of homophily that we have encountered in earlier chapters can sometimes act as a barrier to diffusion: since people tend to interact with others who are like themselves, while new innovations tend to arrive from outside the system, it can be difficult for these innovations to make their way into a tightly-knit social community. With these considerations in mind, we now begin the process of formulating a model for the spread of an innovation through a social network. 19. Modeling Diffusion through a Network We build our model for the diffusion of a new behavior in terms of a more basic, underlying model of individual decision-making: as individuals make decisions based on the choices of their neighbors, a particular pattern of behavior can begin to spread across the links of the network. To formulate such an individual-level model, it is possible to start either
4 566 CHAPTER 19. CASCADING BEHAVIOR IN NETWORKS from informational effects [, 38, 186] or direct-benefit effects [6, 147, 308, 40]. In this chapter, we will focus on the latter, beginning with a natural model of direct-benefit effects in networks due to Stephen Morris . Network models based on direct-benefit effects involve the following underlying consideration: you have certain social network neighbors friends, acquaintances, or colleagues and the benefits to you of adopting a new behavior increase as more and more of these neighbors adopt it. In such a case, simple self-interest will dictate that you should adopt the new behavior once a sufficient proportion of your neighbors have done so. For example, you may find it easier to collaborate with co-workers if you are using compatible technologies; similarly, you may find it easier to engage in social interaction all else being equal with people whose beliefs and opinions are similar to yours. A Networked Coordination Game. These ideas can be captured very naturally using a coordination game, a concept we first encountered in Section 6.5. In an underlying social network, we will study a situation in which each node has a choice between two possible behaviors, labeled A and B. If nodes v and w are linked by an edge, then there is an incentive for them to have their behaviors match. We represent this using a game in which v and w are the players and A and B are the possible strategies. The payoffs are defined as follows: if v and w both adopt behavior A, they each get a payoff of a>0; if they both adopt B, they each get a payoff of b>0; and if they adopt opposite behaviors, they each get a payoff of 0. We can write this in terms of a payoff matrix, as in Figure Of course, it is easy to imagine many more general models for coordination, but for now we are trying to keep things as simple as possible. w A B A a, a 0, 0 v B 0, 0 b, b Figure 19.1: A-B Coordination Game This describes what happens on a single edge of the network; but the point is that each node v is playing a copy of this game with each of its neighbors, and its payoff is the sum of its payoffs in the games played on each edge. Hence v s choice of strategy will be based on the choices made by all of its neighbors, taken together.
5 19.. MODELING DIFFUSION THROUGH A NETWORK 567 A B A B v A B pd neighbors use A (1-p)d neighbors use B B Figure 19.: v must choose between behavior A and behavior B, based on what its neighbors are doing. The basic question faced by v will be the following: suppose that some of its neighbors adopt A, and some adopt B; what should v do in order to maximize its payoff? This clearly depends on the relative number of neighbors doing each, and on the relation between the payoff values a and b. With a little bit of algebra, we can make up a decision rule for v quite easily, as follows. Suppose that a p fraction of v s neighbors have behavior A, and a (1 p) fraction have behavior B; that is, if v has d neighbors, then pd adopt A and (1 p)d adopt B, as shown in Figure 19.. So if v chooses A, it gets a payoff of pda, and if it chooses B, it gets a payoff of (1 p)db. Thus, A is the better choice if or, rearranging terms, if pda (1 p)db, p b a + b. We ll use q to denote this expression on the right-hand side. This inequality describes a very simple threshold rule: it says that if at least a q = b/(a + b) fraction of your neighbors follow behavior A, then you should too. And it makes sense intuitively: when q is small, then A is the much more enticing behavior, and it only takes a small fraction of your neighbors engaging in A for you to do so as well. On the other hand, if q is large, then the opposite holds: B is the attractive behavior, and you need a lot of your friends to engage in A before you switch to A. There is a tie-breaking question when exactly a q fraction of a node s
6 568 CHAPTER 19. CASCADING BEHAVIOR IN NETWORKS r s r s v w v w t u t u (a) The underlying network (b) Two nodes are the initial adopters r s r s v w v w t u t u (c) After one step, two more nodes have adopted (d) After a second step, everyone has adopted Figure 19.3: Starting with v and w as the initial adopters, and payoffs a = 3 and b =, the new behavior A spreads to all nodes in two steps. Nodes adopting A in a given step are drawn with dark borders; nodes adopting B are drawn with light borders. neighbors follow A; in this case, we will adopt the convention that the node chooses A rather than B. Notice that this is in fact a very simple and in particular, myopic model of individual decision-making. Each node is optimally updating its decision based on the immediate consideration of what its neighbors are currently doing, but it is an interesting research question to think about richer models, in which nodes try to incorporate more long-range considerations into their decisions about switching from B to A. Cascading Behavior. In any network, there are two obvious equilibria to this networkwide coordination game: one in which everyone adopts A, and another in which everyone adopts B. Guided by diffusion questions, we want to understand how easy it is, in a given
7 19.. MODELING DIFFUSION THROUGH A NETWORK 569 situation, to tip the network from one of these equilibria to the other. We also want to understand what other intermediate equilibria look like states of coexistence where A is adopted in some parts of the network and B is adopted in others. Specifically, we consider the following type of situation. Suppose that everyone in the network is initially using B as a default behavior. Then, a small set of initial adopters all decide to use A. We will assume that the initial adopters have switched to A for some reason outside the definition of the coordination game they have somehow switched due to a belief in A s superiority, rather than by following payoffs but we ll assume that all other nodes continue to evaluate their payoffs using the coordination game. Given the fact that the initial adopters are now using A, some of their neighbors may decide to switch to A as well, and then some of their neighbors might, and so forth, in a potentially cascading fashion. When does this result in every node in the entire network eventually switching over to A? And when this isn t the result, what causes the spread of A to stop? Clearly the answer will depend on the network structure, the choice of initial adopters, and the value of the threshold q that nodes use for deciding whether to switch to A. The above discussion describes the full model. An initial set of nodes adopts A while everyone else adopts B. Time then runs forward in unit steps; in each step, each node uses the threshold rule to decide whether to switch from B to A. 1 The process stops either when every node has switched to A, or when we reach a step where no node wants to switch, at which point things have stabilized on coexistence between A and B. Let s consider an example of this process using the social network in Figure 19.3(a). Suppose that the coordination game is set up so that a = 3 and b = ; that is, the payoff to nodes interacting using behavior A is 3/ times what it is with behavior B. Using the threshold formula, we see that nodes will switch from B to A if at least a q =/(3 + ) = /5 fraction of their neighbors are using A. Now, suppose that nodes v and w form the set of initial adopters of behavior A, while everyone else uses B. (See Figure 19.3(b), where dark circles denote nodes adopting A and lighter circles denote nodes adopting B.) Then after one step, in which each of the other nodes evaluates its behavior using the threshold rule, nodes r and t will switch to A: for each of them, /3 > /5 of their neighbors are now using A. Nodes s and u do not switch, on the other hand, because for each of them, only 1/3 < /5 of their neighbors are using A. 1 While we won t go through the details here, it is not hard to show that no node that switches to A at some point during this process will ever switch back to B at a later point so what we re studying is indeed a strictly progressive sequence of switches from A to B. Informally, this fact is based on the observation that for any node that switches to A at some point in time, the number of neighbors of this node that follow A only continues to increase as time moves forward beyond this point so if the threshold rule said to switch to A at some point in time, it will only say this more strongly at future times. This is the informal version of the argument, but it is not hard to turn this into a proof.
8 570 CHAPTER 19. CASCADING BEHAVIOR IN NETWORKS Figure 19.4: A larger example. In the next step, however, nodes s and u each have /3 > /5 of their neighbors using A, and so they switch. The process now comes to an end, with everyone in the network using A. Notice how the process really is a chain reaction: nodes v and w aren t able to get s and u to switch by themselves, but once they ve converted r and t, this provides enough leverage. It s also instructive to consider an example in which the adoption of A continues for a while but then stops. Consider the social network in Figure 19.4, and again let s suppose that in the A-B coordination game, we have a = 3 and b =, leading to a threshold of q =/5. If we start from nodes 7 and 8 as initial adopters (Figure 19.5(a)), then in the next three steps we will first see (respectively) nodes 5 and 10 switch to A, then nodes 4 and 9, and then node 6. At this point, no further nodes will be willing to switch, leading to the outcome in Figure 19.5(b). We ll call this chain reaction of switches to A a cascade of adoptions of A, and we d like to distinguish between two fundamental possibilities: (i) that the cascade runs for a while but stops while there are still nodes using B, or (ii) that there is a complete cascade, in which every node in the network switches to A. We introduce the following terminology for referring to the second possibility. Consider a set of initial adopters who start with a new behavior A, while every other node starts with behavior B. Nodes then repeatedly evaluate the decision to switch from B to A using a threshold of q. If the resulting cascade of adoptions
9 19.. MODELING DIFFUSION THROUGH A NETWORK (a) Two nodes are the initial adopters (b) The process ends after three steps Figure 19.5: Starting with nodes 7 and 8 as the initial adopters, the new behavior A spreads to some but not all of the remaining nodes.
10 57 CHAPTER 19. CASCADING BEHAVIOR IN NETWORKS of A eventually causes every node to switch from B to A, then we say that the set of initial adopters causes a complete cascade at threshold q. Cascading Behavior and Viral Marketing. There are a few general observations to note about the larger example in Figure First, it nicely illustrates a point from the opening section, that tightly-knit communities in the network can work to hinder the spread of an innovation. Summarizing the process informally, A was able to spread to a set of nodes where there was sufficiently dense internal connectivity, but it was never able to leap across the shores in the network that separate nodes 8-10 from nodes 11-14, or that separate node 6 from node. As a result, we get coexistence between A and B, with boundaries in the network where the two meet. One can see reflections of this in many instances of diffusion for example, in different dominant political views between adjacent communities. Or, in a more technological setting, consider the ways in which different social-networking sites are dominated by different age groups and lifestyles people will have an incentive to be on the sites their friends are using, even when large parts of the rest of the world are using something else. Similarly, certain industries heavily use Apple Macintosh computers despite the general prevalence of Windows: if most of the people you directly interact with use Apple software, it s in your interest to do so as well, despite the increased difficulty of interoperating with the rest of the world. This discussion also suggests some of the strategies that might be useful if A and B in Figure 19.5 were competing technologies, and the firm producing A wanted to push its adoption past the point at which it has become stuck in Figure 19.5(b). Perhaps the most direct way, when possible, would be for the maker of A to raise the quality of its product slightly. For example, if we change the payoff a in the underlying coordination game from a = 3 to a = 4, then resulting threshold for adopting A drops from q =/5 down to q =1/3. With this threshold, we could check that all nodes would eventually switch to A starting from the situation in Figure 19.5(b). In other words, at this lower threshold, A would be able to break into the other parts of the network that are currently resisting it. This captures an interesting sense in which making an existing innovation slightly more attractive can greatly increase its reach. It also shows that our discussion about the coexistence between A and B along a natural boundary in the network depended not just on the network structure, but also on the relative payoffs of coordinating on A versus B. When it s not possible to raise the quality of A in other words, when the marketer of A can t change the threshold a different strategy for increasing the spread of A would be to convince a small number of key people in the part of the network using B to switch to A, choosing these people carefully so as to get the cascade going again. For example, in Figure 19.5(b), we can check that if the marketer of A were to focus its efforts on convincing node 1 or 13 to switch to A, then the cascading adoption of A would start up again,
11 19.3. CASCADES AND CLUSTERS 573 b f j a d e h i l c g k Figure 19.6: A collection of four-node clusters, each of density /3. eventually causing all of nodes to switch. On the other hand, if the marketer of A spent effort getting node 11 or 14 to switch to A, then it would have no further consequences on the rest of the network; all other nodes using B would still be below their threshold of q =/5 for switching to A. This indicates that the question of how to choose the key nodes to switch to a new product can be subtle, and based intrinsically on their position in the underlying network. Such issues are important in discussions of viral marketing , and have been analyzed in models of the type we are considering here [71, 13, 40, 309, 348]. Finally, it is useful to reflect on some of the contrasts between population-level network effects in technology adoption, as we formulated them in Chapter 17, and network-level cascading adoption as illustrated here. In a population-level model, when everyone is evaluating their adoption decisions based on the fraction of the entire population that is using a particular technology, it can be very hard for a new technology to get started, even when it is an improvement on the status quo. In a network, however, where you only care about what your immediate neighbors are doing, it s possible for a small set of initial adopters to essentially start a long fuse running that eventually spreads the innovation globally. This idea that a new idea is initially propagated at a local level along social network links is something one sees in many settings where an innovation gains eventual widespread acceptance Cascades and Clusters We continue exploring some of the consequences of our simple model of cascading behavior from the previous section: now that we ve seen how cascades form, we look more deeply at what makes them stop. Our specific goal will be to formalize something that is intuitively apparent in Figure 19.5 that the spread of a new behavior can stall when it tries to break in to a tightly-knit community within the network. This will in fact provide a way of formalizing a qualitative principle discussed earlier that homophily can often serve as a barrier to diffusion, by making it hard for innovations to arrive from outside densely connected communities.
12 574 CHAPTER 19. CASCADING BEHAVIOR IN NETWORKS Figure 19.7: Two clusters of density /3 in the network from Figure As a first step, let s think about how to make the idea of a densely connected community precise, so that we can talk about it in the context of our model. A key property of such communities is that when you belong to one, many of your friends also tend to belong. We can take this as the basis of a concrete definition, as follows. We say that a cluster of density p is a set of nodes such that each node in the set has at least a p fraction of its network neighbors in the set. For example, the set of nodes a, b, c, d forms a cluster of density /3 in the network in Figure The sets e, f, g, h and i, j, k, l each form clusters of density /3 as well. As with any formal definition, it s important to notice the ways in which it captures our motivation as well as some of the ways in which it might not. Each node in a cluster does have a prescribed fraction of its friends residing in the cluster as well, implying some level of internal cohesion. On the other hand, our definition does not imply that any two particular nodes in the same cluster necessarily have much in common. For example, in any network, the set of all nodes is always a cluster of density 1 after all, by definition, all your network neighbors reside in the network. Also, if you have two clusters of density p, then the union of these two clusters (i.e. the set of nodes that lie in at least one of them) is also a cluster of density p. These observations are consistent with the notion that clusters
13 19.3. CASCADES AND CLUSTERS 575 in networks can exist simultaneously at many different scales. The Relationship between Clusters and Cascades. The example in Figure 19.7 hints at how the cluster structure of a network might tell us something about the success or failure of a cascade. In this example, we see two communities, each of density /3, in the network from Figure These correspond precisely to the parts of the network that the cascading behavior A was unable to break into, starting from nodes 7 and 8 as initial adopters. Could this be a general principle? In fact it is, at least within the context of the model we ve developed. We now formulate a result saying, essentially, that a cascade comes to a stop when it runs into a dense cluster; and furthermore, that this is the only thing that causes cascades to stop . In other words, clusters are the natural obstacles to cascades. Here is the precise statement, phrased in terms of the set of initial adopters and the remaining network the portion of the network consisting of all nodes other than these initial adopters. Claim: Consider a set of initial adopters of behavior A, with a threshold of q for nodes in the remaining network to adopt behavior A. (i) If the remaining network contains a cluster of density greater than 1 q, then the set of initial adopters will not cause a complete cascade. (ii) Moreover, whenever a set of initial adopters does not cause a complete cascade with threshold q, the remaining network must contain a cluster of density greater than 1 q. It is appealing how this result gives a precise characterization for the success or failure of a cascade, in our simple model, using a natural feature of the network structure. Further, it does so by concretely formalizing a sense in which tightly-knit communities block the spread of cascades. We now prove this result by separately establishing parts (i) and (ii). In going through the proofs of the two parts, it s useful to think about them both in general, and also in light of the example in Figure 19.7, where clusters of density greater than 1 = 3 block the 5 5 spread of A at threshold. 5 We begin with part (i). Part (i): Clusters are Obstacles to Cascades. Consider an arbitrary network in which behavior A is spreading with threshold q, starting from a set of initial adopters. Suppose that the remaining network contains a cluster of density greater than 1 q. We now argue that no node inside the cluster will ever adopt A. Indeed, assume the opposite that some node inside the cluster does eventually adopt A and consider the earliest time step t at which some node inside the cluster does so. Let
14 576 CHAPTER 19. CASCADING BEHAVIOR IN NETWORKS cluster v Figure 19.8: The spread of a new behavior, when nodes have threshold q, stops when it reaches a cluster of density greater than (1 q). v be the name of a node in the cluster that adopts A at time t. The situation is depicted schematically in Figure 19.8 essentially, we want to argue that, at the time v adopted, it could not possibly have had enough neighbors using A to trigger its threshold rule. This contradiction will show that v in fact could not have adopted. Here is how we do this. At the time that v adopted A, its decision was based on the set of nodes who had adopted A by the end of the previous time step, t 1. Since no node in the cluster adopted before v did (that s how we chose v), the only neighbors of v that were using A at the time it decided to switch were outside the cluster. But since the cluster has density greater than 1 q, more than a 1 q fraction of v s neighbors are inside the cluster, and hence less than a q fraction of v s neighbors are outside the cluster. Since these are the only neighbors who could have been using A, and since the threshold rule requires at least a q fraction of neighbors using v, this is a contradiction. Hence our original assumption, that some node in the cluster adopted A at some point in time, must be false. Having established that no node in the cluster ever adopts A, we are done, since this shows that the set of initial adopters does not cause a complete cascade. Part (ii): Clusters are the Only Obstacles to Cascades. We now establish part (ii) of our claim, which says in effect that not only are clusters a natural kind of obstacle to cascades they are in fact the only kind of obstacle. From a methodological point of view (although all the details are different), this is reminiscent of a question we asked with matching markets: having found that constricted sets are natural obstacles to perfect matchings, we went on to find that they are in fact the only obstacle.
15 19.3. CASCADES AND CLUSTERS 577 w nodes that don't eventually switch to A nodes that eventually switch to A initial adopters Figure 19.9: If the spread of A stops before filling out the whole network, the set of nodes that remain with B form a cluster of density greater than 1 q. To prove part (ii) we show that whenever a set of initial adopters fails to cause a complete cascade with threshold q, there is a cluster in the remaining network of density greater than (1 q). In fact, this is not difficult: consider running the process by which A spreads, starting from the initial adopters, until it stops. It stops because there are still nodes using B, but none of the nodes in this set want to switch, as illustrated in Figure Let S denote the set of nodes using B at the end of the process. We want to claim that S is a cluster of density greater than 1 q, which will finish the proof of part (ii). To see why this is true, consider any node w in this set S. Since w doesn t want to switch to A, it must be that the fraction of its neighbors using A is less than q and hence that the fraction of its neighbors using B is greater than 1 q. But the only nodes using B in the whole network belong to the set S, so the fraction of w s neighbors belonging to S is greater than 1 q. Since this holds for all nodes in S, it follows that S is a cluster of density greater than 1 q. This wraps up our analysis of cascades and clusters; the punch-line is that in this model, a set of initial adopters can cause a complete cascade at threshold q if and only if the remaining network contains no cluster of density greater than (1 q). So in this sense, cascades and clusters truly are natural opposites: clusters block the spread of cascades, and whenever a cascade comes to a stop, there s a cluster that can be used to explain why.
16 578 CHAPTER 19. CASCADING BEHAVIOR IN NETWORKS Figure 19.10: The years of first awareness and first adoption for hybrid seed corn in the Ryan-Gross study. (Image from .) 19.4 Diffusion, Thresholds, and the Role of Weak Ties One of the fundamental things we learn from studying diffusion is that there is a crucial difference between learning about a new idea and actually deciding to adopt it. This contrast was already important in the early days of diffusion research. For example, Figure comes from the original Ryan-Gross study of hybrid seed corn ; it shows a clear wave of awareness of this innovation that significantly precedes the wave of adoptions. Our models also illustrate this contrast. If we imagine that people first hear about an innovation when any of their neighbors first adopts, then we see for example in Figure 19.5 that nodes 4 and 9 are aware of A as a new behavior right away, but it takes further time for them to actually adopt it. In an even stronger direction, nodes and eventually become aware of A but never adopt it. Centola and Macy  and Siegel  make the interesting observation that threshold models for diffusion thus highlight an interesting subtlety in the strength-of-weak-ties theory that we discussed in Chapter 3. Recall that the strength of weak ties is rooted in the idea that weak social connections, to people we see infrequently, often form local bridges in a social network. They therefore provide access to sources of information things like new job opportunities that reside in parts of the network we otherwise wouldn t have access to. To take a canonical picture from Chapter 3, shown here in Figure 19.11, the u-w and v-w edges span tightly-knit communities that wouldn t otherwise be able to communicate, and
17 19.4. DIFFUSION, THRESHOLDS, AND THE ROLE OF WEAK TIES 579 u w x v Figure 19.11: The u-w and v-w edges are more likely to act as conduits for information than for high-threshold innovations. thus we expect v, for example, to receive information from his edge to w that he wouldn t get from his other edges. But things look very different if we consider the spread of a new behavior that requires not just awareness, but an actual threshold for adoption. Suppose, for example, w and x in Figure are the initial adopters of a new behavior that is spreading with a threshold of 1/. Then we can check that everyone else in their tightly-knit six-node community will adopt this behavior, but u and v will not. (Nor, therefore, will anyone else lying beyond them in the network.) This illustrates a natural double-edged aspect to bridges and local bridges in a social network: they are powerful ways to convey awareness of new things, but they are weak at transmitting behaviors that are in some way risky or costly to adopt behaviors where you need to see a higher threshold of neighbors doing it before you do it as well. In this sense, nodes u and v in Figure have strong informational advantages over other members of their respective tightly-knit communities they can learn from node w about a new behavior currently spreading in w s community but for behaviors with higher thresholds they will still want to align themselves with others in their own community. If we think about it, this is actually remarkably consistent with the picture from Chapter 3, in which local bridges and positions near structural holes can provide access to information that you re not otherwise
18 580 CHAPTER 19. CASCADING BEHAVIOR IN NETWORKS learning about from your own cluster in the network: for behaviors that spread with high thresholds, a local bridge may well connect you to someone whose network neighborhood has caused them to settle on a different behavior than you have. The trade-offs inherent in this picture have been used to motivate some of the reasons why many social movements tend to build support locally and relatively slowly. Although a world-spanning system of weak ties in the global friendship network is able to spread awareness of a joke or an on-line video with remarkable speed, political mobilization moves more sluggishly, needing to gain momentum within neighborhoods and small communities. Thresholds provide a possible reason: social movements tend to be inherently risky undertakings, and hence individuals tend to have higher thresholds for participating; under such conditions, local bridges that connect very different parts of the network are less useful. Such considerations provide a perspective on other well-known observations about social movements in the diffusion literature, such as Hedstrom s findings that such movements often spread geographically , and McAdam s conclusion that strong ties, rather than weak ties, played the more significant role in recruitment to student activism during Freedom Summer in the 1960s [90, 91] Extensions of the Basic Cascade Model Our discussion thus far has shown how a very simple model of cascades in networks can capture a number of qualitative observations about how new behaviors and innovations diffuse. We now consider how the model can be extended and enriched, keeping its basic points the same while hinting at additional subtleties. Heterogeneous Thresholds. Thus far we have been keeping the underlying model of individual behavior as simple as possible everyone has the same payoffs, and the same intensity of interaction with their network neighbors. But we can easily make these assumptions more general while still preserving the structure of the model and the close connection between cascades and clusters. As the main generalization we consider, suppose that each person in the social network values behaviors A and B differently. Thus, for each node v, we define a payoff a v labeled so that it is specific to v that it receives when it coordinates with someone on behavior A, and we define a payoff b v that it receives when it coordinates with someone on behavior B. When two nodes v and w interact across an edge in the network, they are thus playing the coordination game in Figure Almost all of the previous analysis carries over with only small modifications; we now briefly survey how these changes go. When we first defined the basic coordination game, with all nodes agreeing on how to value A and B, we next asked how a given node v should
19 19.5. EXTENSIONS OF THE BASIC CASCADE MODEL 581 w A B A a v v,a w 0, 0 B 0, 0 b v,b w Figure 19.1: A-B Coordination Game choose its behavior based on what its neighbors are doing. A similar question applies here, leading to a similar calculation. If v has d neighbors, of whom a p fraction have behavior A and a (1 p) fraction have behavior B, then the payoff from choosing A is pda v while the payoff from choosing B is (1 p)db v. Thus A is the better choice if p b v a v + b v. Using q v to denote the right-hand side of this, we again have a very simple decision rule now, each node v has its own personal threshold q v, and it chooses A if at least a q v fraction of its neighbors have done so. Moreover the variation in this set of heterogeneous node thresholds has an intuitive meaning in terms of the variation in payoffs: if a node values A more highly relative to B, its threshold q v is correspondingly lower. The process now runs as before, starting from a set of initial adopters, with each node evaluating its decision according to its own threshold rule in each time step, and switching to A if its threshold is reached. Figure shows an example of this process (where each node s threshold is drawn to the upper-right of the node itself). A number of interesting general observations are suggested by what happens in Figure First, the diversity in node thresholds clearly plays an important role that interacts in complex ways with the structure of the network. For example, despite node 1 s central position, it would not have succeeded in converting anyone at all to A were it not for the extremely low threshold on node 3. This relates closely to a point made in work by Watts and Dodds , who argue that for understanding the spread of behaviors in social networks, we need to take into account not just the power of influential nodes, but also the extent to which these influential nodes have access to easily influenceable people. It is also instructive to look at how the spread of A comes to a stop in Figure 19.13, and to ask whether the notion of clusters as obstacles to cascades can be extended to hold even in the case when thresholds are heterogeneous. In fact, this is possible, by formulating the notion of a cluster in this setting as follows. Given a set of node thresholds, let s say that a blocking cluster in the network is a set of nodes for which each node v has more than a 1 q v fraction of its friends also in the set. (Notice how the notion of cluster density like the notion of thresholds becomes heterogeneous as well: each node has a different requirement for the fraction of friends it needs to have in the cluster.) By a fairly direct adaptation of the analysis from Section 19.3, one can show that a set of initial adopters will cause a complete
20 58 CHAPTER 19. CASCADING BEHAVIOR IN NETWORKS (a) One node is the initial adopter (b) The process ends after four steps Figure 19.13: Starting with node 1 as the unique initial adopter, the new behavior A spreads to some but not all of the remaining nodes.